Normal modes and frequencies from covariances in molecular dynamics or Monte Carlo simulations

We propose a simple method to obtain normal modes (NMs) and their characteristic frequencies from molecular dynamics or Monte Carlo simulations at any temperature. The resulting NM are consistent with the vibrational density of states (DOS) (every feature of the DOS can be attributed to one or few NMs). At low temperatures they coincide with the ones obtained from the Hessian matrix. We define the NMs (rho(i)) by imposing the condition that their velocities be uncorrelated to each other: rho(i)(t)rho(j)(t) proportional, variant delta(ij), where denotes time average and delta(ij) is Kronecker's delta. With this definition the modes are the eigenvectors of the matrix K(ij)(v)=1/2 [i, j=1,...,3N (N being the number of atoms); m are masses and v atomic velocities]. The eigenvalues of K(ij)(v), lambda(i)(v), represent the kinetic energy in each NM. The ratio between the eigenvalues (lambda(i)(v)) and those obtained using positions (lambda(i)(r)), accelerations (lambda(i)(a)) in K(ij)(v) instead of velocities are a very good approximation to the mode frequencies: 2pinu(i) approximately (lambda(i)(v)/lambda(i)(x))((1/2)) approximately (lambda(i)(a)/lambda(i)(x))((1/4)). We demonstrate the new method using with two cases: an isolated water molecule and a crystalline polymer.